RECENT ADVANCES IN SCIENCE 433 



the present time the applications of the theory of aggregates 

 to physics have been but few, though they are of great 

 promise. It is curious to notice that our conception, formed 

 after Dedekind and Cantor, of the continuum, is parallel to the 

 recent progress of physics, which tends more and more to 

 become a theory of discrete molecules, atoms, and electrons. 

 " Even the existence of the matrix or ether in which these are 

 imbedded has become problematical and open to suspicion." 

 Beginning with the paper of 1902 in which Hilbert laid the 

 foundation for geometrical motion, Van Vleck goes on to men- 

 tion the application of ideas in the theory of sets made by 

 Hadamard in 1 898 to geodesies upon surfaces of negative 

 curvature — almost the only application of the theory to differ- 

 ential geometry — the work of Bohl (1909) and Bernstein (191 1 ) 

 on the existence of a " mean motion " in the theory of secular 

 variations, Borel's treatment of the theory of probability, and 

 some work of Poincare, Birkhoff, Brouwers, and others. It is 

 remarkable that Peano's space-filling curve, which is usually 

 considered to be a refinement of exclusively pure mathematics, 

 actually has a practical bearing on some physical work of 

 Boltzmann on the theory of gases and statistical mechanics. 



Robert L. Moore {Trans. Amer. Math. Soc. 191 5, 16, 27) 

 shows that any plane satisfying the axioms given by Veblen 

 in 1904 contains a system of continuous curves such that, with 

 reference to these curves regarded as straight lines, the plane 

 is an ordinary Euclidean plane, or, in other words, is a " number- 

 plane." Moore also (Annals of Math. 191 5, 16, 123) gives 

 a set of eight postulates for the linear continuum in terms of 

 point and limit. This set has a certain similarity with the set 

 proposed by F. Riesz in 1908 ; and postulates for the linear 

 continuum in terms of point and order have been given by 

 Veblen in 1905 and by Huntington in the same year. 



Arithmetic and Theory of Numbers. — The tercentenary of 

 the publication of Napier's Descriptio of 1614 called forth a 

 number of accounts of Napier's invention of logarithms. The 

 earliest seems to be by Prof. G. A. Gibson (Proc. Roy. Phil. Soc. 

 Glasgow, 1 9 14, 3), whose paper is reprinted in the Napier Ter- 

 centenary Handbook reviewed elsewhere in the present number, 

 while the lecture by Prof. E. W. Hobson is also reviewed else- 

 where in this number. Whereas Gibson gave a very full 

 account of Napier's life, his relations with Briggs, and the 



